Formalising and Computing the Fourth Homotopy Group of the \(3\)-Sphere in Cubical Agda

• Published in: arXiv.org
• Main Authors: Ljungström, Axel, Mörtberg, Anders
• Format: Paper
• Language: English
• Published: Ithaca Cornell University Library, arXiv.org 30.04.2024
• Subjects: Computation; Homotopy theory; Questions; Topology
• ISSN: 2331-8422

Brunerie's 2016 PhD thesis contains the first synthetic proof in Homotopy Type Theory (HoTT) of the classical result that the fourth homotopy group of the 3-sphere is \(\mathbb{Z}/2\mathbb{Z}\). The proof is one of the most impressive pieces of synthetic homotopy theory to date and uses a lot of advanced classical algebraic topology rephrased synthetically. Furthermore, the proof is fully constructive and the main result can be reduced to the question of whether a particular "Brunerie number" \(\beta\) can be normalised to \(\pm 2\). The question of whether Brunerie's proof could be formalised in a proof assistant, either by computing this number or by formalising the pen-and-paper proof, has since remained open. In this paper, we present a complete formalisation in Cubical Agda. We do this by modifying Brunerie's proof so that a key technical result, whose proof Brunerie only sketched in his thesis, can be avoided. We also present a formalisation of a new and much simpler proof that \(\beta\) is \(\pm 2\). This formalisation provides us with a sequence of simpler Brunerie numbers, one of which normalises very quickly to \(-2\) in Cubical Agda, resulting in a fully formalised computer-assisted proof that \(\pi_4(\mathbb{S}^3) \cong \mathbb{Z}/2\mathbb{Z}\).